The largest known quartan prime is currently the largest known generalized Fermat prime: The 1353265-digit 145310^262144+1 = (145310^65536)^4+1^4, found by Ricky L Hubbard. - Jens Kruse Andersen, Mar 20 2011

Primes of the form (a^2 + b^2)/2 such that |a^2 - b^2| is a square. - Thomas Ordowski, Feb 22 2017